Moduli stacks of quiver bundles with applications to Higgs bundles
Mahmud Azam, Steven Rayan
TL;DR
The authors develop a unified framework to construct moduli stacks of quiver bundles by indexing diagrams of vector bundles with fixed simplicial shapes over a base. They prove algebraicity for a broad class of bases and recover Nakajima quiver varieties and Higgs-bundle moduli stacks within this setting, providing an intrinsic construction that yields stacks of morphisms and enabling categorification. The work further explores how quiver mutations and mapping-space perspectives relate to Higgs and non-Abelian Hodge theory, and outlines connections to homotopy theory and higher-categorical moduli problems. While largely foundational, the paper sketches concrete pathways to extend these ideas to derived/spatial geometries and to deepen links with non-abelian Hodge theory, representation theory, and homotopical algebra. Overall, it lays the groundwork for a categorified moduli theory that simultaneously unifies quiver varieties, Higgs bundles, and potential homotopical generalizations through internal categories of stacks.
Abstract
We provide a general method for constructing moduli stacks whose points are diagrams of vector bundles over a fixed base, indexed by a fixed simplicial set -- that is, quiver bundles of a fixed shape. We discuss some constraints on the base for these moduli stacks to be Artin and observe that a large class of interesting schemes satisfy these constraints. Using this construction, we recover Nakajima quiver varieties and provide an alternate construction for moduli stacks of Higgs bundles along with a proof of algebraicity following readily from the algebraicity of moduli stacks of quiver bundles. One feature of our approach is that, for each of the moduli stacks we discuss, there are moduli stacks that are Artin, parametrizing morphisms of the objects being classified. We discuss some potential applications of this in categorifying non-abelian Hodge theory in a sense we will make precise. We also discuss potential applications of our methods and perspectives to the subjects of quiver varieties, abstract moduli theory, and homotopy theory.